Simplify; express your answer in exponential form. Assume $n\neq 0, x\neq 0$. $\dfrac{{(nx^{5})^{5}}}{{(n^{-2}x^{2})^{-3}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(nx^{5})^{5} = (n)^{5}(x^{5})^{5}}$ On the left, we have ${n}$ to the exponent ${5}$ . Now ${1 \times 5 = 5}$ , so ${(n)^{5} = n^{5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(nx^{5})^{5}}}{{(n^{-2}x^{2})^{-3}}} = \dfrac{{n^{5}x^{25}}}{{n^{6}x^{-6}}}$ Break up the equation by variable and simplify. $\dfrac{{n^{5}x^{25}}}{{n^{6}x^{-6}}} = \dfrac{{n^{5}}}{{n^{6}}} \cdot \dfrac{{x^{25}}}{{x^{-6}}} = n^{{5} - {6}} \cdot x^{{25} - {(-6)}} = n^{-1}x^{31}$